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Q 1. How many zeroes can the polynomial 2x^{2 } 3x + 4 have?
It is a quadratic polynomial i.e. the degree of 2x^{2 } 3x + 4 is 2.
∴ It can have two zeroes at the most.
Q 2. The graph of y = p (x) is shown in the figure below. How many zeroes does p (x) have?
Since the curve (graph) of p (x) is intersecting the xaxis at three points,
∴ y = p (x) has 3 zeroes.
Q 3. What is the value of p (x) = x^{2}  3x  4 at x = 2?
Given,
p(x) = x^{2}  3x  4
⇒ p(2) = (2)^{2}  3 (2)  4
⇒ 4  6  4 =  6
Q 4. What is the value of p (x) = x^{2}  3x  4 at x =  1?
We have,
p(x) = x^{2 } 3x  4
⇒ p ( 1) = ( 1)^{2 } {3 ( 1)}  4
⇒ 1 + 3  4 = 0
Q 5. What is the zero of 2x + 3?
Since,
The zero of a linear polynomial =
∴ The zero of 2x + 3 = 3/2
Q 6. The Coefficient of x and the constant term in a linear polynomial are 5 and  3 respectively. Find its zero.
Since,
The zero of a linear polynomial
∴ The zero of the given linear polynomial
Q 7. The graph of a quadratic polynomial x^{2}  3x  4 is a parabola. Determine the opening of the parabola.
In x^{2}  3x  4, the Coefficient of x^{2} is 1 and 1 > 0
∴ The parabola opens upwards.
Q 8. Find the sum of the zeroes of the polynomial 2x^{2 } 8x + 6.
p (x) = x^{2}  3x  4
p (x) = 2x^{2}  8x + 6
∴ a = 2, b =  8 and c = 6
⇒ Sum of the zeroes = b/a
⇒ (8)/(2) = 4
Q 9. Find the product of zeroes of the quadratic polynomial 3x^{2} + 5x  2.
Here, a = 3, b = 5 and c =  2
As, Product of the zeroes = c/a
⇒ Product of the zeroes of 3x^{2} + 5x  2 = 2/3
Q 10. If a fifth degree polynomial is divided by a quadratic polynomial, write the possible degree of the quotient.
∵ Degree of the dividend p(x) = 5
Degree of the divisor f(x) = 2
∴ Degree of the quotient q(x) = Degree of p(x)  Degree of f(x) = 5  2 = 3
Q 11. Graph of a linear polynomial is a straight line. If the points (0, 4), (1, 2) and (2, 0) lie on this graph, write the zero of the polynomial.
y = p (x) has its zero at a point where y = 0
In the figure, y = 0 at (2, 0).
∴ 2 is the zero of p (x).
Q 12. For what value of k, ( 4) is a zero of the polynomial x^{2}  x  (2k + 2)?
Since, ( 4) is the zero of p (x)
⇒ p ( 4) = ( 4)^{2}  ( 4)  (2k + 2) = 0
⇒ 16 + 4 = 2k + 2
⇒ 20  2 = 2k
⇒ 18= 2k
⇒ k = 18/2 = 9
Thus, the required value of k = 9.
Q 13. For what value of p, ( 4) is a zero of the polynomial x^{2}  2x  (7p + 3)?
We have
∵ p (x) = x^{2}  2x  (7p + 3)
p ( 4) = ( 4)^{2}  2 ( 4)  (7p + 3)
Since  4 is a zero of p (x),
∴ ( 4)^{2}  2 ( 4)  (7p + 3) = 0
⇒ 16 + 8 = 7p + 3
⇒ 24  3 = 7p
⇒ 7p = 21
⇒ p = 21/7= 3
Thus, the required value of p is 3.
Q 14. The sum and product of the zeroes of a quadratic polynomial are and  3 respectively. What is the quadratic polynomial?
For a quadratic polynomial, we have
Sum of zeroes = 1/2Product of the zeroes = 3
Since general equation of the quadratic polynomial
= x^{2}  [Sum of the roots] x + [Product of the roots]
⇒
Q 15. The graph of y = f (x) is given in the figure, find the number of zeroes of f (x).
From the figure (graph), it is evident that y = f (x) intersects the xaxis at two distinct points A and B.
∴ Number of zeroes of f (x) = 2.
Q 16. In the adjoining figure the graph of a polynomial p (x) is given. Find the zeroes of the polynomial.
From the graph, it is evident that the graph of the given polynomial intersects the xaxis at  1 and  3.
∴ The zeroes of the polynomial are  1 and  3.
Q 17. If 1 is a zero of the polynomial p (x) = ax^{2}  3 (a  1) x  1, then find the value of a.
Here, p (x) = ax^{2}  3(a  1)x  1
∴ p(1) = a(1)^{2 } 3(a  1) × 1  1
= a  3a + 3  1
=  2a + 2
∵ 1 is a zero of p (x);
∴ p (1) = 0
⇒  2a + 2 = 0 ⇒ a = 1
Q 18. If (x + a) is a factor of 2x^{2} + 2ax + 5x + 10 find a.
Here, p(x) = 2x^{2 }+ 2ax + 5x + 10
∵ (x + a) is a factor of p(x)
∴  a is a zero of p(x)
⇒ p ( a) = 0
⇒ 2 ( a)^{2} + 2a ( a) + 5 ( a) + 10 = 0
⇒ 2a^{2}  2a^{2}  5a + 10 = 0
⇒ a = 10/5 = 2
Thus, the required value of a = 2.
Q 19. Write the zeroes of the polynomial x^{2} + 2x + 1.
We have,
p (x) = x^{2} + 2x + 1
= (x + 1)^{2}
If p (x) = 0 then (x + 1)^{2} = 0
⇒ x =  1
Thus, the zero of the polynomial p (x) is  1.
Q 20. Write the zeroes of the polynomial x^{2}  x  6.
We have,
p (x) = x^{2}  x  6
= x^{2}  3x + 2x  6
⇒ x (x  3) + 2 (x  3)
⇒ (x  3) (x + 2)
For p (x) = 0
⇒ (x  3) (x + 2) = 0
∴ x = 3 and x =  2
Q 21. Write a quadratic polynomial, the sum and product of whose zeroes are 3 and  2 respectively.
A quadratic polynomial is given by
x^{2}  [Sum of the zeroes] x + [Product of zeroes]
But, Sum of zeroes = 3
Product of zeroes =  2
∴ The required quadratic polynomial
= x^{2 } 3x + ( 2)
= x^{2}  3x  2
Q 22. Write the number of zeroes of the polynomial y = f (x) whose graph is given in the figure.
Since, curve (graph) of y = f (x) intersects the xaxis at three points,
∴ y = f (x) has three zeroes.
Q 23. The graph of y = f (x) is given in figure. How many zeroes are there of f (x)?
The graph of y = f (x) does not intersect the xaxis.
∴ The number of zeroes of f (x) is 0.
Q 24. The graph of y = f (x) is given in the figure. What is the number of zeroes of f (x)?
Since, the graph of y = f (x) intersects the xaxis at three distinct points,
∴ y = f (x) has 3 zeroes.
Q 25. What is the number of zeroes of the polynomial y = p (x)?
Since, the graph passes through origin(0,0) which means on putting x = 0, we get y = 0.
Therefore, (0,0) is the only root of this equation.
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